An electron (spin ) of mass moves in the isotropic oscillator

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Symmetry, angular momentum and spin · Central potentials and canonical three-dimensional systems

The Structure of Quantum Mechanics, Ch Central potentials as a rotation problem

Problem

An electron (spin s=12s=\tfrac12) of mass mm moves in the isotropic oscillator V(r)=12mω2r2V(r)=\tfrac12m\omega^2r^2. Its level N=2N=2 holds the orbital multiplets l=2l=2 and l=0l=0, twelve states once spin is counted. Now add a coupling A L⃗^⋅S⃗^A\,\hat{\vec L}\cdot\hat{\vec S} with a constant A>0A\gt0. (a) Explain why L⃗^⋅S⃗^\hat{\vec L}\cdot\hat{\vec S} commutes with the oscillator Hamiltonian and with L^2\hat L^2, so that the states of the N=2N=2 level can still be labelled by ll, jj and mjm_j. (b) Using L⃗^⋅S⃗^=12(J^2−L^2−S^2)\hat{\vec L}\cdot\hat{\vec S}=\tfrac12(\hat J^2-\hat L^2-\hat S^2), find the shift of each sublevel in units of Aℏ2A\hbar^2 and its degeneracy. (c) Check that the degeneracy-weighted shifts sum to zero, and say which degeneracy of the uncoupled level the coupling destroys and which it keeps.

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